The utility function is: I In(C2) U = In(C1) + 1+P The budget constraint is: -C2 =Y1+ T+r 1 Y2 C1 + 1+r 1. Find the utility maximizing values for C, and C2
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- What kind of preferences are represented by a utility function of theform u(x1, x2) = √x1 + x2? What about the utility function v(x1, x2) =13x1 + 13x2?What is the logarithmic transform of the utility function U=xα1xβ2xγ3U=x1αx2βx3γ given the budget constraint px1x1+px2x2+px3x3=Mpx1x1+px2x2+px3x3=M. Select one: a. ln U=ln xα1+ln xβ2+ln xγ3ln U=ln x1α+ln x2β+ln x3γ b. ln U=α ln x1−β ln x2−γ ln x3ln U=α ln x1−β ln x2−γ ln x3 c. ln U=α ln x1+β ln x2+γ ln x3ln U=α ln x1+β ln x2+γ ln x3 d. ln U=ln xγ1+ln xβ2+ln xα3There are 6 cups of espresso (x1) and 6 bars of chocolate (x2) in the house. Andrew and Eric have identical preferences, U₁ = alog(x) + (1 - a)log(x), where i = A, E and a € (0, 1). Suppose the Target allocation is ((2, 2), (4, 4)). (a) What is the relationship between UA and UE in the Target? (b) Find the ratio of equilibrium prices at the Target allocation. (c) Determine the lump-sum transfer necessary to achieve the Target if An- drew is initially endowed with (4, 1) and Eric with (2,5). Set x2 as the numeraire, i.e., assume pi 1. Under what condition would require Andrew to transfer to Eric or vice versa? Why? If a > 1/3, Andrew transfers to Eric and vice versa. = p and p2 =
- For the utility function U = (Qx0.5+Qy0.5)2 and the budget 190 = 5Qx + 12Qy find the CHANGE in optimal consumption of Y if the price of X increases by a factor of 1.8. Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 = 1.2, 1.14 = 1.1). The answer is 2.1 but how?Problem 3 Consider a CES utility function. 0-1 0-1\0-1 Uf = + a2 (c2t) Derive u; as 0 → 1.2) Which of the following utility functions represent the same preferences? Explain. a) U (x₁, x₂) = X₁ X₂ b) W (x₁, x₂) = 5lnx₁ +5lnx₂ c) V (x₁, x₂) = x₁¹/3x₂ ¹/3 - 0.8 d) Z(x₁, x₂) = 0.5x₁ + 0.5x₂
- Find Indirect utility function of the following function U = max (X, Y) subject to the budget constraint P, X+ P₂ Y=M Optionis O O max (P1 P2) M² min (P1 P2) M² P1+ P2 M min (P1 P2)Anthony seeks to maximize the following utility function u(x, y) = x'/3y2/3 subject to the budget constraint Pæa + PyY = I 1 where pr, Py, x, y, I > 0. a) Find Anthony's utility-maximizing bundle (x*, y*) as a function of pæ, Py, and I. b) Show that y* is decreasing in py and increasing in I (hint: use partial derivatives). c) What share of Anthony's income is spent on x? What share is spent on y? In other words, calculate Pa and Pu. Are these shares a function of prices? Pyy* Note: The above utility function is Cobb-Douglas, and all Cobb-Douglas functions have these share formulas for any values for the exponents. d) What is the impact of a change in pr on Anthony's utility?Donald likes fishing (X1) and hanging out in his hammock (X2). His utility function for these two activities is u(x1, x2) = 3X12X24. (A) Calculate MU1, the marginal utility of fishing. (B) Calculate MU2, the marginal utility of hanging out in his hammock. (C) Calculate MRS, the rate at which he is willing to substitute hanging out in his hammock for fishing. (D)Last week, Donald fished 2 hours a day, and hung out in his hammock 4 hours a day. Using your formula for MRS from (c) find his MRS last week. (E) This week, Donald is fishing eight hours a day, and hanging out in his ham mock two hours a day. Calculate his MRS this week. Has his MRS increased or decreased? Explain why? (F) Is Donald happier or sadder this week compared to last week? Explain.
- For the following utility function find the MRS at (x, y) = (2, 3) a) U(x, y) =2x +y b) U(x, y) =x^3 y^2 c) U(x, y) =min(x, 2y) Solve subparts (c )correctlySummer break is approaching! Suppose you derive utility from days spent traveling on vacation domestically, D, and days spent traveling on vacation in a foreign country, F. Your utility function over these two “goods” is: U(D, F) : 4D0.25F 0.75 Let your budget constraint be I(D, F) = pDD + pF F; where I is your income, pD is the price of domestic travel per day, and pF is the price of foreign travel per day. a. Determine the demand functions for domestic travel and foreign travel. Make sure you show your work – show the steps used. b. Suppose that you’ve saved $800 for your summer travel, the price of domestic travel per day is $25 and the price of foreign travel per day is $100. How many days of each type of travel will you embark on? c. Illustrate the indifference curve, budget constraint, and the utility maximizing bundle associated with (b). Make sure you show the level of utility, the budget constraint intercepts, and the optimizing equilibrium.Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3
